In this section we outline the basic features of the two most common models with genuine long-memory properties.
6.3.1 Fractional Gaussian Noise
Definition A real-valued stochastic process X ={Z(t)}t∈R is self-similar with index
H >0 (H-self-similar) if, for any a >0
where =ddenotes equality in distribution and Rdenotes the real numbers. The index H is called the self-similarity parameter, the scaling exponent or the Hurst parameter in honour of H. E. Hurst who brought the phenomena to prominence and developed the R/S statistic (see Section 7.2 below).
Definition A real-valued process{Z(t)}t∈Rhas stationary increments if, for allh∈ R, {Z(t+h)−Z(h)}t∈R =d{Z(t)−Z(0)}t∈R.
H-self-similar processes with stationary increments (H-sssi) give rise to stationary sequences with long memory. For a time series{Xj}, if {Z(t)}t∈R is anH-sssi process then
Xj = ∆Z(j) =Z(j+ 1)−Z(j) , j∈ Z (6.7) whereZ denotes the integers, is a stationary sequence.
Definition A GaussianH-sssi process{BH(t)}t∈R with 0< H <1 is called fractional Brownian motion (FBM). It is called standard ifσ2= var[BH(1)] = 1.
Definition If {Z(t)}t∈R is an FBM, then the increment of the process (Equation 6.7) is called fractional Gaussian noise (FGN).
Taqqu (2003) states that the increment sequence {Xk}k∈Z has the following prop- erties:
1. {Xk}k∈Z is stationary. 2. E[Xk] = 0.
3. E[Xk2] =σ2 =E[Z(1)2].
4. The autocovariance function of the process{Xk}k∈Z is given by
γ(k) =E[XiXi+k] = σ 2 2 |k+ 1| 2H −2|k|2H+|k−1|2H = σ 2 2 ∆ 2|k|2H
where ∆2 denotes the second difference. 5. Fork6= 0 γ(k) = <0,0< H <12 = 0, H= 12 >0,12 < H <1. 6. IfH 6= 1/2, then γ(k)≈σ2H(2H−1)|k|2H−2 ask→ ∞ (6.8) From Equation (6.8) γ(k) tends to zero like a power function with increasing k. When 1/2< H <1 it tends to zero so slowly that the sum
∞
X
k=−∞
γ(k)
diverges. Thus the process represented by Equation (6.7) displays long-memory. It can be shown (Beran, 1994, p53) that the spectral density of FGN is
f(ω)∼cω1−2H
where ω is the frequency and c is a constant. Thus the long-memory property cor- responds to a divergence of the spectral representation to infinity at the origin. Of course, in finite samplesf(0) is finite.
Some authors were enthusiastic about the ability of FGNs to model hydrological time series. Wallis and O’Connell (1973) wrote “It has been exhaustively documented that discrete time approximations to fractional Gaussian noise provides a necessary and sufficient explanation of the Hurst phenomenon”. On the other hand, Hipel and McLeod (1978) reported that FGNs were inferior to several other processes in modeling long memory series, in particular to the Box-Jenkins ARMA models.
6.3.2 ARFIMA Models
The other common true long memory model is the extension of Box-Jenkins ARIMA(p,d,q) models to non-integer values of d which was accomplished independently by Granger
and Joyeux (1980) and Hosking (1981). Several factors motivated these authors to consider fractional differencing.
For a stationary short memory time series the sample ACF decays exponentially as the lag increases. For a non-stationary random walk or unit-root process the sample ACF converges to one for all fixed lags as the sample size increases. For series with a unit-root, taking first differences yields a stationary time series with an ACF which decays exponentially as before. Intermediate between these are long memory time series which have a sample ACF which decays at a polynomial rate with increasing lag and for which taking first differences yields a series which appears to be over-differenced. Thus some model with an order of differencing intermediate between zero and one seemed required.
Hosking (1981) attributed the above observation of Hipel and McLeod (1978) to FGNs’ inability to model low-lag correlation structures correctly and stated this moti- vated his research when he added non-zero AR and MA orders to his fractional differ- encing model.
These ARIMA(p,d,q) models with non-integer dvalues are usually called AutoRe- gressive Fractionally Integrated Moving Average (ARFIMA) series and are normally defined for −1/2 < d < 1/2 as any other series can be differenced until d lies in this range. When −1/2 < d <0 the series is anti-persistent. Henry and Zaffaroni (2003) state that anti-persistent series are characterized by a shrinking spectral density to- wards zero frequency.
Recall (Chatfield, 2004, p48) that an ARIMA(p,d,q) model is defined as
φ(B)(1−B)dXt=θ(B)t (6.9) whereB is the backward shift operator (B[Xt] =Xt−1), andφ(B) and θ(B) are poly- nomials in B of order p and q respectively, Xt is the observation at timet and the t
are usually assumed to be a white noise sequence drawn from anN(0,1) distribution. If, for simplicity, we assume φ(B) =θ(B) = 1 then Equation (6.9) reduces to
or
Xt= (1−B)−dt. (6.11) If −1/2 < d < 1/2 then Equation (6.10) can be written as an infinite order AR process (Beran, 1994, pp64,65) ∞ X k=0 πkXt−k=t (6.12) where πk= Γ(k−d) Γ(k+ 1)Γ(−d) and Γ(·) is the gamma function.
Similarly, Equation (6.11) can be written as an infinite order MA process
Xt= ∞ X k=0 a(k)t−k (6.13) where a(k) = Γ(k+d) Γ(k+ 1)Γ(d).
It can be shown (Beran, 1994, pp63-64) that the covariance function is given by
γ(k) =σ2 (−1)
kΓ(1−2d) Γ(k−d+ 1)Γ(1−k−d) and the correlations are equal to
ρ(k) = Γ(1−d)Γ(k+d) Γ(d)Γ(k+ 1−d), and ρ(k)∼ Γ(1−d) Γ(d) |k| 2d−1 for|k| → ∞.
Thus for 0 < d < 1/2 a fractionally integrated process exhibits long memory as all previous states of the process influence the present.
In the last paragraph of Hosking (1981) he mentions in passing that fractional differencing may prove useful with the process (1−2φB+B2)dxt=t, with |d|<0.5 and |φ| < 1. This bears a striking resemblance to the so-called Gegenbauer process which is elaborated on in Beran (1994, pp213–215). Gegenbauer processes exhibit both long memory and behaviour which is approximately cyclic but not strictly periodic.